Double Folium

Cartesian equation:
(x² + y²)² = 4axy²
Polar equation:
r = 4a cosθ sin²θ

Click below to see one of the Associated curves.

Definitions of the Associated curves	Evolute
Involute 1	Involute 2
Inverse curve wrt origin	Inverse wrt another circle
Pedal curve wrt origin	Pedal wrt another point
Negative pedal curve wrt origin	Negative pedal wrt another point
Caustic wrt horizontal rays	Caustic curve wrt another point

(x²+y²)(y²+x(x+b)) = 4axy²

or, in polar coordinates

r = -b cosθ + 4a cosθ sin²θ.

The word folium means leaf-shaped.

There are three special forms of the folium, the simple folium, the double folium and the trifolium. These correspond to the cases

b = 4a, b = 0, b = a

respectively in the formula for the general form.

The graph plotted above is the double folium. There are separate entries for the simple folium and the trifolium.

The double folium is the pedal curve of the tricuspoid where the pedal point is mid-point of one of the three curved sides.

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Double.html